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A Guide to linear dynamic analysis with Damping This guide starts from applications of linear dynamic response and its role in FEA simulation. Fundamental concepts and principles will be introduced such as equation of motion, types of vibrations, role of damping in engineering and its types, linear dynamic analyses, etc. Finally we will discuss how to choose appropriate solution type for damping and introduce strength of midas NFX for solving dynamic problems.
Courtesy of Faculty of Mechanical Engineering and Mechatronics; West Pomeranian University of Technology, Poland
1. Dynamic Analysis Application Dynamic analysis is strongly related to vibrations.
Vibrations are generally defined as fluctuations of a mechanical or structural system about an equilibrium position. Vibrations are initiated when an inertia element is displaced from its equilibrium position due to an energy imported to the system through an external source. Vibrations occur in many mechanical and structural systems. Without being controlled, vibrations can lead to catastrophic situations. Vibrations of machine tools or machine tool chatter can lead to improper machining of parts. Structural failure can occur because of large dynamic stresses developed during earthquakes or even wind induced vibrations. Figure 1: 1940 Tacoma Narrows Bridge failure Vibrations induced by an unbalanced helicopter blade while rotating at high speeds can lead to the blade's failure and catastrophe for the helicopter. Excessive vibrations of pumps, compressors, turbomachinery, and other industrial machines can induce vibrations of the surrounding structure, leading to inefficient operation of the machines while the noise produced can cause human discomfort.
Figure 2: Failed compressor components
Vibrations as the science is one of the first courses where most engineers to apply the knowledge obtained from mathematics and basic engineering science courses to solve practical problems. Solution of practical problems in vibrations requires modeling of physical systems. A system is abstracted from its surroundings. Usually assumptions appropriate to the system are made. Basic engineering science, mathematics and numerical methods are applied to derive a computer based model. Page 2
2. Dynamic Analysis Equation The mathematical modeling of a physical system results in the formulation of a mathematical problem. The modelling is not complete until the appropriate mathematics is applied and a solution obtained. The type of mathematics required is different for different types of problems. Modeling of any statics, dynamics, and mechanics of solids problems leads only to algebraic equations. Mathematical modeling of vibrations problems leads to differential equations. In mathematical physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.
Equations of motion are consisted of inertial force, damping force (energy dissipation) and elastic (restoring) force. The overall behavior of a structure can be grasped through these three forces.
mu(t ) INERTIA FORCE
cu (t )
DAMPING FORCE
ku(t )
RESTORING FORCE
p(t ) APPLIED FORCE
Inertia Force is generated by accelerated mass. Damping Force describes energy dissipation mechanism which induces a force that is a function of a dissipation constant and the velocity. This force is known as the general viscous damping force. The final induced force in the dynamic system is due to the elastic resistance in the system and is a function of the displacement and stiffness of the system. This force is called the elastic force, restoring force or occasionally the spring force. The applied load has been presented on the right-hand side of equation and is defined as a function of time. This load is independent of the structure to which it is applied. Exact analytical solutions, when they exist, are preferable to numerical or approximate solutions. Exact solutions are available for many linear problems, but for only a few nonlinear problems. Page 3
2.1 Single Degree of Freedom System Simple mechanical system is schematically shown in Figure 3. The inputs (or excitation) applied to the system are represented by the force p(t). The outputs (or response) of the system are represented by the displacement u(t). The system boundary (real or imaginary) demarcates the region of interest in the analysis. What is outside the system boundary is the environment in which the system operates. Environment
System Boundary
Dynamic System State of Variables Parameters (m, c, k)
System Excitation (Inputs)
u(t)
System Response (Outputs)
Figure 3: A Mechanical dynamic system
System parameters are represented in the model, and their values should be known in order to determine the response of the system to a particular excitation.
State variables are a minimum set of variables, which completely represent the dynamic state of a system at any given time t. For a simple SDOF oscillator an appropriate set of state variables would be the displacement u and the velocity du/dt. The equation of motion for SDOF mechanical system may be derived using the free-body diagram approach.
2.2 Equation of Motion for Single Degree of Freedom System
p(t ) mu(t ) cu (t ) ku(t ) m – mass c – damping coefficient k – stiffness coefficient p(t) – applied force
Figure 4: SDOF free body diagram. Page 4
u(t)– mass displacement u(t)– mass velocity u(t)– mass acceleration
2.3 Single Degree of Freedom System Responses Free Undamped Response General solution:
mu(t ) ku(t ) 0 u (t )
u 0
n
sin n t u0 cos n t
u(t ) A cos n t B sin n t Natural frequency: n
k rad [ ] m s
u0 and u0 are an Initial Condition entities Free Damped Response General solution:
mu(t ) cu(t ) ku(t ) 0 c ccr 2 km 2mn
Critically Damped System
u(t ) ( A Bt )e ct / 2 m
Underdamped solution: u(t ) e ct / 2 m ( A sin d t B cos d t ) c ccr
c ccr
Overdamped System
Underdamped System
d n 1 2
Damped natural frequency
c ccr
Damping ratio
Forced Undamped Response
Forced Damped Response
(t ) ku(t ) p(t ) mu
mu(t ) cu(t ) ku(t ) p(t )
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3. Dynamic Analysis Types in midas NFX 3.1 Eigenvalue Analysis/Normal Modes/Modal Analysis of the Normal Modes or Natural Frequencies of a structure is a search for it’s resonant frequencies. By understanding the dynamic characteristics of a structure experiencing oscillation or periodic loads, we can prevents resonance and damage of the structure.
Mass
Stiffness
Structure
Natural Frequencies Normal Mode Shapes
Constraints
Natural Frequency – the actual measure of frequency, [Hz] or similar units Normal Mode shape – the characteristic deflected shape of a structure as it resonates
3.2 Transient Analysis Executed in the time domain, it obtains the solution of a dynamic equation of equilibrium when a dynamic load is being applied to a structure. Though the load and boundary conditions required for a transient response analysis are similar to those of a static analysis, a difference is that load is defined as a function of time.
Mass
Stiffness
Response in Time Domain
Structure
Constraints
Damping
Load in Time Domain
3.3 Frequency Response Analysis A technique to determine the steady state response of a structure according to sinusoidal ( harmonic) loads of known frequency. Frequency Response is best visualized as a response to a structure on a shaker table. Adjusting the frequency input to the table gives a range of responses.
Mass
Stiffness
Structure
Constraints
Damping Response in Freq. Domain
Load in Freq. Domain
mu(t ) cu (t ) ku(t ) F sin(t )
u (t ) F / k
sin(t ) 2 (1 -
n
2
)2 ( 2 / n )2
input or driving frequency phase lead
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Figure 5: SDOF frequency response - Displacement
Figure 6: Shaking table for frequency response experiment
4. Dynamic Loading Types Dynamic loads can be applied directly on your model or as time or frequency dependent static loadings.
- Time domain based loading types
- Frequency domain based loading types
Figure 7. Domain dependent Static Loading.
4.1 Solution Methods Dynamic Analysis
Normal Modal Analysis (Eigenvalue Analysis)
Direct Integration Method
Linear or Nonlinear Analysis Modal Superposition Method
Linear Analysis
Transient Response Analysis
Frequency Response Analysis
Shock and Response Analysis* *Not Covered in this guide
Figure 8. Dynamic Analysis Solution Types and Methods. Page 7
5. Damping Damping is the phenomenon by which mechanical energy is dissipated (usually by conversion into internal thermal energy) in dynamic systems. Knowledge of the level of damping in a dynamic system is important in the utilization, analysis, and testing of the system. In structural systems, damping is more complex, appearing in several forms.
Internal damping refers to the structural material itself. Internal (material) damping results from mechanical energy dissipation within the material due to various microscopic and macroscopic processes.
Mechanical Damping
- Internal - External
- Distributed - Localized
Figure 9. Damping forms External damping comes from boundary effects. An important form is structural damping, which is produced by rubbing friction: stick-and-slip contact or impact. Another form of external damping is fluid damping.
Figure 10. Localized damping examples
All damping ultimately comes from frictional effects, which may however take place at different scales. If the effects are distributed over volumes or surfaces at macro scales, we speak of distributed damping. Damping devices designed to produce beneficial damping effects, such as shock absorbers, represent localized damping.
5.1 Damping Models It is not practical to incorporate detailed microscopic representations of damping in the dynamic analysis of systems. Instead, simplified models of damping that are representative of various types of energy dissipation are typically used. Damping Models
General Viscous Damping Modal damping Rayleigh damping
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Structural / Hysteretic
Coulomb
Figure 11. General Damping models review
Fluid
5.2 Damping Models in midas NFX The following types of damping are available inside the midas NFX: - Proportional Damping (Rayleigh; classic) - Hysteretic/Structural Damping - Direct Damping values - Frequency dependent damping - Modal Damping - Coulomb damping, requires special modelling techniques
5.3 Damping Model input for: - Rayleigh Damping (Proportional)
- Discrete Viscous Damping
- Defined via Material Card
- BUSH 1D element Property - Damper element Property
- Structural Damping
- Defined via Material Card and Analysis Case Manager - Overall and Elemental Damping
- Modal Damping
- Coulomb Damping
- Defined via Analysis Case Manager and Functions
- Defined by combination of elements and features
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6. Project Applications of Dynamic Analysis in midas NFX midas NFX are used in various projects to investigate dynamic characteristics of a large range of industrial products. Some of the examples are shown below. Case 1
Performance evaluation of a mobile speaker through sound pressure level (SPL) analysis
Mode Frequency
(midas NFX application from KEYRIN TELECOM)
In this project, the performance of a cellphone speaker is evaluated under different sound pressure levels.
1,778 Hz
3ND
1,801 Hz
4ND
9,457 Hz
5ND
9,679 Hz
Displacement / Frequency
Case 2
Resonance avoidance of ultra large AC servo robot (midas NFX application from YUDO STAR Automation)
Mode
Natural Frequency
1st Mode
6.0Hz
2nd Mode
8.2Hz
3rd Mode
14.6Hz
Resonance frequency range
From the image we can confirm the necessity for resonance avoidance design in order to avoid the resonance which happens at 15Hz under repetitive load.
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2ND
Stress
From the right image, we can see different mode shapes of the structure. We can also observe that maximum displacement was 0.4 mm, it occurred at around 1000Hz. At this frequency the suspension structure reached its maximum stress of 250MPa.
Stresses were equally calculated under seismic load through response spectrum analysis.
1,046 Hz
6ND 10,217 Hz
Firstly modal analysis was performed to determine natural frequencies of the speaker components. Then frequency response analysis was performed to calculate stresses and deformation shapes of the speaker components according to different frequency spectrums.
In this case a dynamic characteristics of a servo robot arm was investigated both to avoid resonance during machine operation and to ensure structural safety during earthquakes.
1ND
Stress distribution under seismic load
Case 3
Brake Disc Squeal Analysis (midas NFX application)
In this project the dynamic characteristics of brake disc were reviewed to avoid squeal noise caused by vibration. Through frequency response analysis, we can observe that at around 7000 Hz, frequencies of Nodal Diameter Mode and In-Plane Compression Mode are very close, where squeal noise is most likely to occur. Therefore, a design modification is needed to separate 2 frequencies to avoid squealing problems. Natural
Mode
frequency
2ND
1027 Hz
3ND
2394 Hz
4ND
3897 Hz
5ND
5468 Hz
1st IPC
7014 Hz
6ND
7045 Hz
7ND
8592 Hz
8ND
10068 Hz
2nd IPC
10285 Hz
1st IPC
6ND
Case 4
Safety analysis of marine refrigeration machine under vibration (midas NFX application)
In this project, natural frequency analysis and frequency response analysis were performed to predict the happening of cracks on the body and piping of a marine refrigeration machine under vibration loads. 100
Pipe① Frequency Part frequency Response Function response X-direction
Y-direction
10
Z-direction
1
100
Pipe②Frequency Part frequency Response Function response X-direction
Y-direction
10
Z-direction
1
0.1
0.1
1
0.01
0.01
0.001
0.001
0.0001 0
10
20
30
40
50 60 Frequency
3
70
80
2
90
100
0.0001 0
100
10
20
30
40
50 60 Frequency
70
80
90
100
Pipe③Frequency Part frequency Response Function response X-direction
Y-direction
10
Z-direction
1 0.1 0.01
Resonant stress distribution
0.001
0.0001
Resonant displacement distribution
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0
10
20
30
40
50 60 Frequency
70
80
90
100
7. Advantages of NFX for Dynamic Analysis 7.1 Easiness to be learned and adopted Despite all the benefits that simulation can bring to product design, adopting “simulation driven design” approach in a well established team can be sometimes challenging. To fully leverage the capabilities of FEA simulation and create maximum values, design engineers need to learn FEA software as well as some specialized engineering knowledge. Sometime design engineers need to work together with analysis specialist in the team to complete a more complex analysis. midas NFX is well-known for its fast learning curve to both analysis specialists and product designers. midas NFX divides its interface into Designer Mode and Analyst Mode to fit the needs of simple and powerful analysis in one software. Designer mode aims to help design engineers who aspire to take that intellectual leap from CAD design to FEA simulation. It is equipped with automation tools such as automatic simplification, auto-meshing, analysis wizards, etc., as well as powerful meshing algorithms and highperformance solvers.
Analyst mode provides experienced analyst full flexibility to build and analyze finite element models in the way you want. One can use different types of elements for modeling, and also generate mesh manually. There are also tools to create and edit geometry that can help the user tweak with the geometry without the inconvenience of returning to the CAD platform. With the “auto-update” function, user can create “ analysis template” with which when model is redesigned, analysis can be directly performed on the model without having to assign all the analysis condition again and again. Besides time saving, you can also extend knowledge of the analysis specialist by asking a specialist to create the template so that the design engineers can use it later in the “design – simulation” iterations.
7.2 Easiness for modeling New product are designed in CAD tools, many CAD tools are available on the market, and even in one team different CAD tools are usually being used. Therefore, CAD compatibility is another critical capacity to consider. midas NFX is fully compatible with CAD, 13 most commonly used CAD formats are supported, including Parasolid, CATIA V4/V5, UG, Pro/E, SolidWorks, Soild Edge, Inventor, STEP, IGES, ACIS. Nastran format (.nas; .bdf) can also be import for analysis.
Models directly taken from CAD designs are usually with excessive detailing. Especially electronic components, machine parts include many small holes, fillets, lines, faces which are not only unnecessary but also will lead to inefficient analysis. Common practice is to clean up the geometry model first before meshing and analyzing it.
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midas NFX is equipped with automatic and manual simplification tools to handle effectively the clean up work. With several clicks such detailing can be automatically detected, selected and removed. This will save you considerably amount of time. Furthermore, midas NFX is equipped with geometry creation and editing tools. Simple modifications to the model can be directly done in midas NFX without going back to CAD tools.
7.3 Convenient graphical output and reporting Dynamic analysis provides the output related to the nature of the problem: Natural vibration analysis can extract natural vibration frequencies and the associated mode shapes
The response analysis refers to the calculation of the response, which can be displacements, strain, or stress, when the system is subjected to time or frequency varying excitation forces A good software provides result visualization in the most appropriate manner, as well provides tools to extract flexibly any result at any location of interest. Advanced requirements such as drawing value tables and graphs should also be satisfied. Result visualization is one of the biggest strengths of midas NFX. The software provides different kinds of contour maps. With “ISO Surface”, overheated locations can be easily discovered. Interactive graph for incremental solution provides an excellent possibility to investigate and validate your model during solution phase.
midas NFX has the provision for automatic report generation. The reports include all the information related to the model, right from the geometry to the result graphs and tables. The automatic reports can be generated in MS-Word or 3D PDF format. The former can use default or custom templates whereas the latter is an interactive report with animated illustrations of the model in a 3D view.
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